The goal is to defend falsificationism (as formulated my Karl Popper) from this criticism:
Theories that contain existential statements, such as "some mammals lay eggs," are not falsifiable (because one cannot observe the whole universe in order to falsify the statement), but the one I mentioned above seems perfectly scientific.

I already tried saying this: "falsificationists need to bite the bullet and accept that a theory like that is unscientific, but also say that few, if not none of, theories are of that form anyway (the concept of electron specifies a location where it can be observed, and the theory of universal gravitation applies to all location and time)." But my professor said that there is a way to defend that does not involve biting the bullet.

As I understand it, Popper's falsificationism doesn't apply to every statement you could possibly derive from a scientific theory. It applies to theories as a whole, which contain hypotheses that propose universals.

An individual statement that is falsifiable but not verifiable can easily be turned into a statement that is verifiable but not falsifiable by negating it. e.g. "all swans are white" is falsifiable but not verifiable. Its negation "some swans are not white" is the reverse. It's verifiable but not falsifiable.

In your case, the statement "some mammals lay eggs" is verifiable. Its negation is the statement "there are no mammals that lay eggs". That statement is falsifiable (by finding at least one egg-laying mammal).

"When the seagulls follow the trawler, it is because they think sardines will be thrown into the sea." - Eric Cantona.